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%I #33 Nov 08 2022 08:07:58
%S 1,16383,2391484,134209536,1525878906,39179682372,113037178808,
%T 1099444518912,3812797945332,24998474116998,37974983358324,
%U 320959957991424,328114698808274,1851888100411464,3649114989636504
%N a(n) = Sum_{d|n} Moebius(n/d)*d^(b-1)/phi(n) for b = 15.
%C a(n) is the number of lattices L in Z^14 such that the quotient group Z^14 / L is C_n. - _Álvar Ibeas_, Nov 26 2015
%H Enrique Pérez Herrero, <a href="/A161025/b161025.txt">Table of n, a(n) for n = 1..5000</a>
%H Jin Ho Kwak and Jaeun Lee, <a href="https://doi.org/10.1142/9789812799890_0005">Enumeration of graph coverings, surface branched coverings and related group theory</a>, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161. See p. 134.
%H <a href="/index/J#nome">Index to Jordan function ratios J_k/J_1</a>.
%F a(n) = J_14(n)/J_1(n) where J_14 and J_1(n) = A000010(n) are Jordan functions. - _R. J. Mathar_, Jul 12 2011
%F From _Álvar Ibeas_, Nov 26 2015: (Start)
%F Multiplicative with a(p^e) = p^(13e-13) * (p^14-1) / (p-1).
%F For squarefree n, a(n) = A000203(n^13). (End)
%F From _Amiram Eldar_, Nov 08 2022: (Start)
%F Sum_{k=1..n} a(k) ~ c * n^14, where c = (1/14) * Product_{p prime} (1 + (p^13-1)/((p-1)*p^14)) = 0.1388226555... .
%F Sum_{k>=1} 1/a(k) = zeta(13)*zeta(14) * Product_{p prime} (1 - 2/p^14 + 1/p^27) = 1.00006146517418... . (End)
%p A161025 := proc(n)
%p add(numtheory[mobius](n/d)*d^14,d=numtheory[divisors](n)) ;
%p %/numtheory[phi](n) ;
%p end proc:
%p for n from 1 to 5000 do
%p printf("%d %d\n",n,A161025(n)) ;
%p end do: # _R. J. Mathar_, Mar 15 2016
%t A161025[n_]:=DivisorSum[n,MoebiusMu[n/#]*#^(15-1)/EulerPhi[n]&]
%t f[p_, e_] := p^(13*e - 13) * (p^14-1) / (p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* _Amiram Eldar_, Nov 08 2022 *)
%o (PARI) vector(100, n, sumdiv(n^13, d, if(ispower(d, 14), moebius(sqrtnint(d, 14))*sigma(n^13/d), 0))) \\ _Altug Alkan_, Nov 26 2015
%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^14 - 1)*f[i,1]^(13*f[i,2] - 13)/(f[i,1] - 1));} \\ _Amiram Eldar_, Nov 08 2022
%Y Column 14 of A263950.
%Y Cf. A000010, A000203, A013671, A013672.
%K nonn,mult
%O 1,2
%A _N. J. A. Sloane_, Nov 19 2009
%E Definition corrected by _Enrique Pérez Herrero_, Oct 30 2010
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